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PL _|_ ST wanted in cyclic ABCD

Source: 2022 2nd OMpD L3 P4 - Brazil - Olimpíada Matemáticos por Diversão

October 13, 2022
perpendicularcyclic quadrilateralgeometry

Problem Statement

Let ABCDABCD be a cyclic quadrilateral and M,NM,N be the midpoints of ABAB, CDCD respectively. The diagonals ACAC and BDBD intersect at LL. Suppose that the circumcircle of LMNLMN, with center TT, intersects the circumcircle of ABCDABCD at two distinct points X,YX,Y. If the line MNMN intersects the line XYXY at SS and the line XMXM intersects the line YNYN at PP, prove that PLPL is perpendicular to STST.
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